Animated Solution for Chemistry - Periodicity in Properties: The correct order of ionic radii for the ions, ${\text{P}}^{3-},{\text{S}}^{2-},{\text{Ca}}^{2+},{\text{K}}^{+},{\text{Cl}}^{-}$ is

The problem of comparing ionic radii is a classic favorite in chemistry. It tests your fundamental understanding of atomic structure and the delicate balance of forces within an atom. Let's dive into the fascinating world of isoelectronic species and see how a simple tug-of-war determines their size.

When we first look at the given ions—${\text{P}}^{3-}$, ${\text{S}}^{2-}$, ${\text{Cl}}^{-}$, ${\text{K}}^{+}$, and ${\text{Ca}}^{2+}$—they might seem like a random assortment from different parts of the periodic table. Some are non-metals that have gained electrons, while others are metals that have lost electrons.

However, the secret to solving this lies in finding their common ground. Let's count the total number of electrons for each ion. Phosphorus normally has $15$ electrons, but the $3-$ charge means it has gained $3$, giving it $18$ electrons. Sulfur ($16$) gains $2$ to reach $18$. Chlorine ($17$) gains $1$ to reach $18$. Potassium ($19$) loses $1$ to drop to $18$, and Calcium ($20$) loses $2$ to also reach $18$.

They all have exactly $18$ electrons! In chemistry, we call such a group isoelectronic species.

Effective Nuclear Charge

Since every single ion in this lineup has the exact same number of electrons, the electron-electron repulsion is roughly the same across the board. So, what makes their sizes different? The answer lies in the nucleus.

The size of an ion is determined by a tug-of-war between the positively charged protons in the nucleus pulling inward, and the negatively charged electrons pushing outward. Because the number of electrons is tied at $18$, the deciding factor is the number of protons, also known as the atomic number ($Z$).

This brings us to the concept of effective nuclear charge ($Z_{\text{eff}}$). For isoelectronic species, the effective nuclear charge is directly proportional to the actual nuclear charge. The more protons you have, the stronger the inward pull on that identical cloud of $18$ electrons.

Let's line up our ions based on their proton count: - Phosphorus (${\text{P}}^{3-}$): $15$ protons - Sulfur (${\text{S}}^{2-}$): $16$ protons - Chlorine (${\text{Cl}}^{-}$): $17$ protons - Potassium (${\text{K}}^{+}$): $19$ protons - Calcium (${\text{Ca}}^{2+}$): $20$ protons

Calcium has a massive team of $20$ protons pulling on its $18$ electrons. This overwhelming positive charge yanks the electron cloud tightly inward, making ${\text{Ca}}^{2+}$ the smallest ion in the group.

On the other extreme, Phosphorus only has $15$ protons trying to hold onto the same $18$ electrons. The nucleus is simply outmatched, allowing the electron cloud to expand outward, making ${\text{P}}^{3-}$ the largest.

Therefore, as the atomic number increases, the ionic radius strictly decreases. The correct decreasing order of their sizes is: ${\text{P}}^{3-}>{\text{S}}^{2-}>{\text{Cl}}^{-}>{\text{K}}^{+}>{\text{Ca}}^{2+}$

This perfectly matches option (a). Always remember: when electrons are tied, the protons dictate the size!